In this paper, we propose a new geometric-shaping design for golden angle modulation (GAM) based on the complex geometric properties of open symmetrized bidisc, termed Bd-GAM, for future generation wireless communication systems. Inspired from the circular symmetric structure of the GAM, we construct the modulation schemes, Bd-GAM1 and Bd-GAM2. However, the optimization problem is hard to solve analytically, we consider this problem in the domain of symmetrized bidisc. Specifically, we joint the complex geometrics properties of symmetric bidisc and MI-optimized probabilistic modulation scheme. The symmetric bidisc is a bounded domain on which the Carathe´odory distance and the Kobayashi distance coincide. We can use the pseudo-metric mD in unit disc D to get the Kobayashi pseudo-distance and Carathe´odory pseudo-distance. With minimum SNR and entropy constraint, Bd-GAM1 and Bd-GAM2 can overcome the shaping-loss. This study finds that each constellation point of Bd-GAM has a unique phase and gain. Compared with the existed golden angle modulation introduced, the new design improves the mutual informaiton (MI), and the distance between adjacent constellation points.